Hyperkähler cohomology and BPS cohomology

I will speak about the relationship between the Hodge numbers of the known Hyperkähler manifolds and the BPS cohomology for the category of coherent sheaves on K3 surfaces. I hope to touch on many of the themes that have already appeared in the seminar: BPS sheaves and CoHAs for 2CY categories, chi-independence, and compactified Jacobians. If I have time and the audience has the energy, I will talk about LLV Lie algebras and a completely unexplored idea to apply them to BPS cohomology.